Essential singular point
An isolated singular point of single-valued character of an analytic function
of a complex variable
at which the limit
, whether finite or infinite, does not exist. In a sufficiently small punctured neighbourhood
of an essential singular point
, or
in case
, the function
can be expanded into a Laurent series:
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or, correspondingly,
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where in the principal part of these series there is an infinite number of non-zero coefficients with negative indices
.
The Sokhotskii theorem asserts that every complex value in the extended complex plane
is a limit value for the function
in any neighbourhood, however small, of an essential singular point
. According to the Picard theorem, every finite complex value
, with one possible exception, is a value of
taken infinitely often in any neighbourhood of an essential singular point
. The Sokhotskii theorem can also be expressed in another way, by stating that the cluster set
of a function
at an essential singular point
coincides with the extended complex plane
. For regular points and poles, this set, on the other hand, is degenerate, i.e. it reduces to a single point
. Therefore, in a more general sense, the name essential singular point of an analytic function
is applied to every singular point
(not necessarily isolated) at which no finite or infinite limit
exists, or, in other words, at which the cluster set
is non-degenerate. The theorems of Sokhotskii and Picard for such essential singular points, not being isolated points of the set of all singular points, have only been proved under certain additional assumptions. For example, these theorems still hold for an isolated point
of the set of essential singular points, in particular for a limit point
of the poles of a meromorphic function.
A point of the complex space
,
, is called a point of meromorphy of an analytic function
of several complex variables
if
is a meromorphic function in a neighbourhood
of
, i.e. if
can be represented in
as a quotient of two holomorphic functions:
,
. Singular points
of
that are not points of meromorphy are called essential singular points of
. In these cases the non-degeneracy of the cluster set
ceases to be a characteristic property of essential singular points.
References
[1] | A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) |
[2] | B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian) |
Comments
In Western literature the Sokhotskii theorem is known as the Casorati–Weierstrass theorem.
References
[a1] | S. Saks, A. Zygmund, "Analytic functions" , Elsevier (1971) (Translated from Polish) |
Essential singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Essential_singular_point&oldid=14859